Analysis and Design of RC Chimney 120m

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REPORT

ON

B.TECH PROJECT

ANALYSIS AND DESIGN OF REINFORCED CONCRETE

CHIMNEYS

BY

N. RAVI KIRAN

CE 97062

UNDER THE GUIDANCE OF

DR DEVDAS MENON

DEPARTMENT OF CIVIL ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY, MADRAS

MAY 2001


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Certificate

This is to certify that the report titled ANALYSIS AND DESIGN OF

REINFORCED CONCRETE CHIMNEYS, submitted by Nagavarapu

Ravi Kiran, to the Indian Institute of Technology, Madras, in partial

fulfillment of the requirements for the award of the degree of Bachelor ofTechnology in Civil Engineering is a bona fide record of the work done by

him under the guidance of Prof. Devdas Menon during the academic year

2000-2001

Dr. Devdas Menon

Project Guide

Associate Professor

Dept. Of Civil EngineeringIIT Madras.

Dr. V.Kalyanaraman

Professor and Head

Dept. Of Civil EngineeringIIT Madras.

Department of Civil Engineering,

Indian Institute of Technology, Madras.


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Acknowledgements

I would like to thank my project guide Dr. Devdas Menon who has been

extremely patient with me during the last one year and without whose help and guidance

this project would not have been possible. I am very indebted to him.

I would like to place on record my thanks to all the faculty of IIT Madras who

have been extremely cooperative and helpful during my stay at the Institute.

I would also like to thank all my class mates, friends and wing mates who have

made my stay at this place a wonderful experience.

N. Ravi Kiran

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Abstract

The present thesis deals with the analysis and design aspects of Reinforced

Concrete Chimneys. The thesis reviews the various load effects that are incident upon tall

free standing structures such as a chimney and the methods for estimation of the same

using various codal provisions. Various loads are incident upon a chimney such as, wind

loads, seismic loads and temperature loads etc. The codal provisions for the evaluation of

the same have been studied and applied. Comparison has also been done between the

values obtained of these load effects using the procedures outlined by various codes.

The design strength of the chimney cross sections has also been estimated. Design

charts have also been prepared that can be used to ease the process of the design of the

chimney cross sections and the usage exemplified.

A typical chimney of 250m has been analyzed and designed using the processes

already outlined. Drawings have been prepared for the chimney. The foundation for the

chimney too has been designed.

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Contents

Title Pno

Chapter 1: Introduction 1

Chapter 2: Estimation of Wind Load Effects 2

2.1 Along Wind Effects 2

2.1.1 Basic Design wind speed 3

2.1.2 Wind Profile 3

2.1.3 Design Wind pressure 5

2.1.4 Force Resultants 6

2.1.5 Dynamic Effects and Gust factor 9

2.1.6 Analysis using STRAP 10

2.1.7 Expected Maximum Moments 11

2.2 Across Wind Effects 12

2.2.1 Vortex Shedding 13

2.2.2 Chimney modeling and estimation of shape factorand time period

14

2.2.3 Estimation of Moments 14

2.2.4 Variation of Moments with change in H/D ratio 20

2.2.5 Conclusions of the variational Analysis 21

2.3 Conclusions 24

Chapter 3: Estimation of Earthquake load Effects 25

3.1 Introduction 25

3.2 Estimation of loads 26

3.2.1 Design seismic coefficients 28

3.3 Calculations for a typical case 29

3.4 Conclusions 32

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Chapter 4: Estimation of Temperature load Effects 33

4.1 Introduction 33

4.2 Equations for evaluation of stresses 354.3 Conclusions 39

Chapter 5: Estimation of Design Resistance and Development of Design

Charts40

5.1 Introduction 40

5.2 Characteristic Stress-Strain Curve for Steel 41

5.3 Characteristic Stress-Strain Curve for Concrete 42

5.4 Calculation of Ultimate Moments 44

5.5 Interaction Curve 46

5.5.1 Family of Interaction Curves 48

5.5.2 Derivation of Equations used 50

5.6 Conclusions 52

Chapter 6: Design and detailing of Example Chimney 53

6.1 Introduction 53

6.2 Design of chimney 53

6.3 Design of foundation 57

6.4 Conclusions 60

Chapter 7: Summary and Conclusion 61

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Acknowledgements i

Abstract iiContents iii

List of figures vi

List of tables vii

List of important symbols viii

Appendix I

References XI

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List of figures

Figure Description

2.1 Wind Profile and Response

2.2 Moment Profiles (comparative)

2.3 Across wind Effect

2.4 Mode shapes

2.5 IS Simplified method & Figure

2.6 IS Random Response Method

2.7 IS Approximate Method Mode 1

2.8 IS Approximate Method Mode 2

2.9 IS Random Response Method Mode 1

2.10 IS Random Response Method Mode 2

3.1 Shear force due to seismic loads

3.2 Bending Moment due to seismic loads

4.1 Thermal Stresses

5.1 Stress-strain curve (steel)5.2 Stress-strain curve (concrete)

5.3 Chimney Cross-section

5.4 Stress and strain distributions

5.5 Strain profile variation

5.6 Interaction Curves

6.1 The Chimney

6.2 Sectional plan view Vertical reinforcement

6.3 Sectional elevation view Horizontal reinforcement

6.4 The foundation (representation)

6.5 Load and eccentricity

6.6 Actual loading pattern

6.7 The foundation and the connection

6.8 Design of staircase tread

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List of Tables

Table Description

2.1 Chimney Attributes

2.2 Results of dynamic analysis

2.3 Base Gust factors (comparative)

2.4 Base Moments (ACI)

2.5 ACI method (all moments)

4.1 Vertical Stresses

4.2 Hoop Stresses

5.1 Values of the interaction curve parameter

6.1 List of chimney parameters used

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List of important symbols used

Symbol Description

Vb , Vh Basic wind speed

Vz Wind profile

CD Drag Coefficient

wm Wind loading

G Gust factor

pz Pressure profile with height

g Peak factor, Acceleration due to gravity

Sn Strouhal number

oi Peak deflection due to vortex shedding in ith mode

Normalized mode shape

Fzoi Force due to vortex shedding

Mzoi Moment due to vortex shedding

Vcr Critical Velocity of flowui Normalized response

ui Actual response

h Seismic Coefficient

I Importance factor

Soil Coefficient

Sa Seismic acceleration

Es

Modulus of Elasticity (Steel)

Ec Modulus of Elasticity (Concrete)

Tx Temperature gradient

st Thermal stress in steel

ct Thermal stress in concrete

Coefficient of thermal expansion

k Location of neutral axis

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sml Limiting stress in steel

cul Limiting stress in concrete

s Partial safety factor for steelc Partial safety factor for concrete

r Radius of the chimney

t Thickness of the chimney shell

m Dimensionless quantity for moment

n Dimensionless quantity for normal force

Percentage of steel

fs

Stress-strain curve for steel

fpc Stress-strain curve for steel

Strain

0 Location of the neutral axis

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Chapter 1. Introduction

This project deals with the analysis and design of Reinforced Concrete (RC)

chimneys. Such chimneys (with heights up to 400m) are presently designed in conformity

with various codes of practice (IS 4998, ACI 307, CICIND etc.). The main loads to be

considered during the analysis of tall structures such as chimneys are wind loads,

temperature loads and seismic loads in addition to the dead loads. The design is done

using limit state concepts (which are yet to be incorporated into IS 4998).

The wind load effects are of two distinct types along-wind effects and across-

wind effects. While the along-wind loads deal with the effect of direct action of the wind

on the face of the chimney, the across-wind loads deals with the aerodynamic action of

the bluff body cross section of the chimney in a wind flow. The evaluation of along-wind

is straight forward, while the across-wind load estimation is more involved requiring

dynamic analysis. The loads are idealized as those on a acting on a cantilever, for the

purpose of evaluation of the load resultants on the chimney.

The seismic loads are another cause of natural loads on the chimney. These loads,

caused by earthquakes are generally dynamic in nature. However the codes provide forquasi-static methods for the evaluation of these loads. Codal provisions normally

recommend amplification of the normalized response of the chimney with a factor that

depends on the local soil conditions and the intensity of the earthquake.

The temperature load effects too are an important consideration in the analysis of

loads effects on chimneys taking into consideration the fact that the chimneys are used

for the venting of hot gasses. This develops a temperature gradient with respect the

ambient temperature outside and hence causes stresses in the reinforced concrete shell.

There is a considerable difference between the methods employed and the

assumptions made by the various codes. Hence the values predicted by the various codes

too vary a lot. A comparison has also been done between the values reported by the

various codes for the wind load effect analysis.

The design of the chimneys requires the estimation of the resistance of the tubular

cross section of the chimney. Also suitable design charts were constructed to serve as

design aids.

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Chapter 2. Estimation of Wind Load Effects

Wind forms the predominant source of loads, in tall freestanding structures like

chimneys. The effect of wind on these tall structures can be divided into two

components, known respectively as

x along-wind effect

x across-wind effect

Along-wind loads are caused by the drag component of the wind force on the

chimney, whereas the across-wind loads are caused by the corresponding lift

component. The former is accompanied by gust buffetting causing a dynamic response

in the direction of the mean flow, whereas the latter is associated with the phenomenon

of vortex shedding which causes the chimney to oscillate in a direction perpendicular

to the direction of wind flow. Estimation of wind effects therefore involves the

estimation of these two types of loads.

2.1 Along Wind Effects

Along-wind effect is due to the direct buffeting action, when the wind acts on the

face of a structure. For the purpose of estimation of these loads the chimney is modeled

as a cantilever, fixed to the ground. The wind is then modeled to act on the exposed face

of the chimney causing predominant moments in the chimney. Additional complications

arise from the fact that the wind does not generally blow at a fixed rate. Wind generally

blows as gusts. This requires that the corresponding loads, and hence the response be

taken as dynamic. True evaluation of the along-wind loads involves modeling the

concerned chimney as a bluff body having incident turbulent wind flow. However, the

mathematical rigor involved in such an analysis is not acceptable to practicing

engineers. Hence most codes use an equivalent static procedure known as the gust

factor method. This method is immensely popular and is currently specified in a number

of building codes including the IS (IS:4998) code. This process broadly involves the

determining of the wind pressure that acts on the chimney due to the bearing on the face

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of the chimney, a static wind load. This is then amplified using the gust factor to take

care of the dynamic effects.

This study involves the evaluation of the along-wind loads by using the methodsspecified in a number of codes like

x CICIND (Model Code for Concrete Chimneys, 1998)

x ACI 307-95

x IS 4998 (Part 1) : 1992

2.1.1 Basic Design Wind speed

One of the primary steps to finding the along-wind loads is to get the basic

design wind speed. The determination of the effective wind pressure is based on the

basic wind speed. The basic wind speed (Vb) is defined (by the CICIND code) as the

mean hourly wind speed at 10m above the ground level in open flat country without

having any obstructions. This means that the wind speed is measured at a height of 10m

above the ground at the location of the chimney and is averaged over an hour. The ACI

code suggests a wind speed averaged over a period of the order of 20min to 1hr. The IS

code however uses the basic wind speed based on peak gust velocity averaged over a

short time interval of about three seconds. The value of the basic wind speed must be

established by meteorological measurement. Normally though it is not necessary to

actually do the measurement for a particular region. The values as suggested from

published Wind Maps specified by the codes may be used. Basic wind speeds generally

have been worked out for a return period of 50 yrs.

It may me noted that the ACI follows the FPS system and therefore in the

following discussion the formulae by the code appear different from the SI system of the

other two codes.

2.1.2 Wind Profile

Wind flow is retarded by frictional contact with the earths surface. The effect of

this retardation is diffused by turbulence in wind flow across a region known as the

atmospheric boundary layer. The thickness of this boundary layer depends on the wind

speed, terrain roughness and angle of latitude. The rougher the terrain, the more

effective the retardation to the mean flow, and hence, greater is the gradient height. The

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effect of this gradient is the wind flow now assumes a profile that varies with height

from 0 at the surface to the maximum at the end of the atmospheric boundary layer.

The variation of mean wind speed with height Vz is generally described by thepower law.

(2.1)Vz = Vb (Z / Zo)

Where Vz is the profile with respect to height. Vb is the basic wind speed, Z is

the height above ground level, Zo) is a height of the boundary layer and is the terrain

factor. The values of the various factors are specified by the respective codes.

The CICIND code suggests the following code for the purpose of evaluation of

the wind speed profile.

(2.2)V(z) = Vb k(z) kt ki

Where:

V(z) is the hourly mean wind speed at level z

z is the height above ground level

Vb is the basic wind speed specified

k(z) is given by the equation

(2.3)k(z) = ks (z / 10)

ks scale factor, equal to 1.0 in open flat country

is the terrain factor

kt topographical factor

ki interference factor

The ACI code gives the following formula for obtaining the Wind profiles

V(z) = (1.47)0.78(80/VR)0.09 VR(z/33)

0.14(2.4)

Where VR is the basic wind speed. The equation also converts from the basic

wind speed in mph to ft/s as required for the calculations.

The IS:875 however does not give a wind profile but gives a wind velocity at any

height Vz.(2.5)

Vz = Vb k1 k2 k3

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Where Vz is the required wind speed, Vb is the basic wind speed. k1 is aprobability factor (risk), k2 is the terrain, height and structure size factor,k3 is a

topography factor. The values of these factors can be gauged from the Tables given in

the IS code.

2.1.3 Design Wind Pressure

The obtained wind velocities are assumed to act on the face of the chimney. The

corresponding pressure on the surface has to be evaluated next. This is done with the

help of the drag coefficient. This coefficient is defined in a number of ways in all the

codes. The main concept however is that the square of the velocity acting at any point is

to be multiplied by this coefficient to get the pressure acting at that point. The

coefficient takes into account factors like slenderness of the column, ribbed quality of

the surface, the effect of having a curved surface etc.

The wind pressure then is multiplied with the density of air and the exposed area

to get the actual static loads acting on the chimney.

The CICIND code calculates the loads with the following formula

wm(z) = 0.5 a v(z)2 CD d(z) (2.6)

Which is more than just the pressure calculation. However the term CD refers to

the coefficient that depends on the slenderness of the column. The value of this

coefficient depends on the h/d ratio and can be obtained from the code. It varies between

0.6 and 0.7 for change in the h/d ratio from 5 to 25. The term wm(z) is basically the

weight acting on the cantilever for which it has to be designed.

The Indian code converts the velocity profile into its corresponding pressure

profile with the help of the following formula

pz = 0.6 Vz2

(2.7)

The value of 0.6 is the drag coefficient specified.

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The ACI code suggests a very similar function, however specifying the

coefficient to be 0.0013 as opposed to 0.6, mainly to keep it consisting with the FPS

system used by the code.

6

Figure 2.1 Wind profile and Response

2.1.4 Force resultants

The pressure values obtained in the earlier case are then converted into the

corresponding force values. The chimney is idealized to be a vertical cantilever, fixed to

the ground. The load that acts can be takes as a continuous load acting on this cantilever.

The calculation of the force resultants of shear and moment are trivial.

In reality the base of the chimney is broad. Hence the shear resisting capacity of

the chimney is high. In fact shear also may manifest itself as moment due to the deep

beam effect. Hence the more important resultant to calculate here is the moment as

compared to either the shear or the axial force.

Moment

Wind

Profile


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The moment at any point on the cantilever can be calculated by integrating the

moment from the end to that point. Hence the functions given to calculate the moment

too are integrals.The CICIND code gives the following main formula for the purpose of

calculation of the gust factor moments in chimneys

h

mg zdzzwh

zGzw

0

2)(

)1(3)( (2.8)

where

G is the gust factor (will be looked into later)

h is the height of the top of the shell above the ground levelz is the height above the ground level

wm(z) is the mean hourly wind load per unit height at height z

The IS code gives two methods for the evaluation of along-wind loads on

chimneys, both of which are discussed below.

The IS simplified method

This method, as the name suggests, is a simple procedure to come up with the

load values for a given configuration. The formula suggested for this method is

Fz = pz.CD.dz (2.9)

Where the factor CD is to be taken as 0.8. This is actually a vast simplification of

the procedure outlined in the IS:875 which specifies the distribution of the value of the

drag coefficient around the periphery of the cylindrical shell. This method however does

not take into account the effect of the dynamic quality of the incident wind on the

chimney.

The second method given by the code is the random response method. The

equations for the same are given below and terms explained. The need and use of the

Gust factor however is discussed later.

H

zmzf zdzFH

z

H

gF

0

2

)1(3(2.10)

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8

8

Where Fzf is the wind load in N/m height due to the fluctuating component of the

wind at height z. The whole load is given by

(2.11)zfzmz FFF

The wind load due to the hourly mean wind component is given by

where pz gives the design pressure at hourly mean wind component and is

pbtained by the equation

zDzzm dCpF (2.12)

zVpz2

6.0 (2.13)

In the equation for the fluctuating component of the wind load the gust factor G

is used. The equations and the concept involved are discussed later.

The ACI code gives the following code for the purpose of calculation of the

along-wind load. This code too divides the load due to the wind into two parts the

mean load and the fluctuating component. The mean load is calculated by the formula

)()()()( zpzdzCzw dr (2.14)Where the value

Cdr= 0.65 for z < h-1.5d(h)

Cdr= 1 for z > h-1.5d(h)

And the value of the mean pressure has been given.

The fluctuating load component has been taken equal to

3

' )(0.3)('h

bMzGzw ww (2.15)

Where M is the base bending moment due to the constant load acting on the

chimney. It is basically an integral of the weight acting on the chimney multiplied with

the distance from the base. The Gust factor G is calculated by

> @86.0

47.0

1

)16(

)33(0.1130.0'

h

VTGw

(2.16)

For a preliminary design the Time period of oscillation can be calculated with

the help of an equation suggested by the code. However the code requires the time


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period to be calculated with the help of dynamic analysis for the final design. Analysis

here was done by modeling the chimney using a program STRAP.

2.1.5 Dynamic Effects and the Gust Factor

All along-wind loads that act on the chimney are not due to the static wing

bearing on the surface of the chimney alone. There is a significant change in the applied

load due to the inherent fluctuations in the strength of wind that acts on the chimney. It

is not possible of feasible to take the maximum load that can ever occur due to wind

loads and design the chimney for the same. At the same time it is very difficult to

quantify the dynamic effect of the load that is incident on the chimney. Such a process

would be very tedious and time consuming. So most of the codes make use of the gust

factor to account for this dynamic loading. To simplify the incident load due to the mean

wind is calculated and the result is amplified by means of a gust factor to take care of the

dynamic nature of the loading.

The gust factor is defined as the ratio of the expected maximum moment M0 to

the mean moment Mm0 at the base of the chimney. It is accordingly denoted as G0 and is

referred to as the base gust factor.

The CICIND code gives the following formula for the calculation of the Gust

factor.

]ES

BgiG 21 (2.17)

Where g is peak factor with

vTvTg

e

elog2

577.0log2 (2.18)

the turbulence intensity

hi 10log089.0311.0 (2.19)

88.063.0

2651

h

Bbackground turbulence(2.20)

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83.0

42.0

2

1

21.01

3301

123

hV

f

hV

f

E

b

benergy density

(2.21)spectrum

88.0

98.0

14.1

178.51

h

V

fS

b

size reduction factor (2.22)

damping is a fraction of the critical damping and is taken as 0.016. f1 is the

natural frequency in the first mode of vibration.

h is the height of the shell above the ground in m and Vb is the basic wind speed.

T is the sample period and v is effective cycling rate.

The equation for the Gust factor used by the ACI code is given earlier.

The IS code probably borrowed its gust factor equation from the CICIND code

as both the equations are remarkably similar. Only the names given to some of the

factors are different. The factors and the equations themselves are the same

A typical chimney of 250m was chosen to calculate the along-wind loads. The

dynamic analysis was done using a structural analysis program called STRAP.

2.1.6 Analysis using STRAP

For the purpose of analysis the chimney was modeled in STRAP. The chimney

was idealized into 32 components outside the ground and one component inside the

ground (to take care of fixity and the effect of the foundation), a total of 33 components.

The various components were taken to be cylindrical objects. Hence the chimney was

idealized as 33 hollow cylinders stacked upon each other.

The thickness of the components of the chimney were varied according the

thickness of the actual chimney at the middle of each section. A fixed joint was assumed

after 32 nodes.

For the purpose of dynamic analysis the weight data was calculated by the

program itself. This however was strictly not correct because there would be the

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additional weight of the lining inside the chimney. Hence a lining of a layer of bricks

was assumed and the weight calculated by the program was corrected with a factor to

account for the weight of the lining. The calculation of the factors was done with thehelp of a small program that actually calculated the volume ratios for the purpose.

The chimney itself was assumed to be of a standard dimensions and ratios as

given below.

Attribute Value

Height 250m

Height to Base Diameter 7

Top Diameter to Base Diameter 0.6Base Diameter to base thickness 35

Top thickness to base thickness 0.4675

Table 2.1 Chimney Attributes

The results of dynamic analysis of the modeled chimney are given below

Mode Time Period

1 0.23452 1.0266

3 2.4826

4 3.6286

5 4.4460

Table 2.2 Results of dynamic analysis

These values of time periods of oscillations and the corresponding frequencies

(1/Time Period) were used for the calculations of the Gust factor.

2.1.7 Expected maximum moments

The moments were calculated for the model chimney assumed earlier and the

results are shown in the graph below

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0

50

100

150

200

250

300

0 500 1000 1500 2000

CICIND ACI IS

Figure 2.2 Moment profiles (comparative)

As is visible, there is considerable difference in the expected maximum base

moments of the chimney using the three codal methods.

Additionally the base gust factors for the three methods are given below

Code Base Gust factor

IS 1.85

CICIND 1.85

ACI 1.993

Table 2.3 Base Gust factors (comparative)

2.2 Across Wind Effects

Recommendations for considering the across-wind loads have been included into

the codes only recently. In spite of considerable research the problem of accurately

predicting the across-wind response has to be fully resolved. Hence the CICIND code

does not take into account across-winds. For this study the codes used therefore were the

IS 4998(Part 1): 1992 and the ACI 307-95.

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A tall body like the chimney is essentially a bluff body as opposed to a

streamlines one. The streamlined body causes the oncoming wind flow to go smoothly

past it and hence is not exposed to any extra forces. On the other hand the bluff bodycauses the wind to separate from the body. This separated flow causes high negative

regions in the wake region behind the chimney. The wake region is a highly turbulent

region that give rise to high speed eddies called vortices. These discrete vortices are

shed alternately giving rise to lift forces that act in a direction perpendicular to the

incident wind direction.

13

Figure 2.3 Across wind effect

These lift forces cause the chimney to oscillate in a direction perpendicular to the

wind flow.

2.2.1 Vortex Shedding

The phenomena of alternately shedding the vortices formed in the wake region is

called vortex shedding. This is the phenomena that gives rise to the across-wind forces.

This phenomena was reported by Strouhal, who showed that shedding from a

circular cylinder in a laminar flow is describable in terms a non-dimensional number S n

called the Strouhal number.

CHIMNEY

velocityflowmean

cylinderofdiameterfrequencysheddingSn

__

___ u (2.23)


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The phenomena of vortex shedding and hence the across-wind loads depends on

a number of factors including wind velocity, taper factors etc., that are specified by the

codes. Codal estimation of the across-wind loads also involves the estimation of themode-shape of the chimney in various modes of vibration. This is obtained as follows.

2.2.2 Chimney Modeling and estimation of shape factor and time period

As discussed earlier dynamic analysis of the chimney was done using the

structural analysis program STRAP. A model chimney with the parameters shown

earlier was modeled and dynamic analysis performed on it. The required mode shapes

were obtained from the program itself.

The results from the analysis are given below with the normalized mode shapes

on the left and the corresponding frequencies of vibration on the right. It may be noted

that although four mode shapes have been assumed for the purpose of analysis, in reality

only the first two modes are actually active. This is because the wind velocity required

to make the chimney vibrate in higher mode shapes is very high.

Mode shapes 1 to 4

Frequencies:

Mode 1: 0.2345 hz

Mode 2: 1.0266 hz

Mode 3: 2.4826 hz

Mode 4: 3.6286 hz0

5

10

15

20

25

30

35

-1.2 -0.7 -0.2 0.3 0.8

Figure 2.4 Mode shapes

2.2.3 Estimation of Moments

The various codal methods for the purpose of estimation of along-wind loads are

as follows.

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The IS code, gives two methods for the estimation of across-wind loads. These

are called respectively the simplified method and the random response method. The

amplitude of the vortex excited oscillation is to be calculated by the equation.

sin

L

H

zi

H

ziz

oiKS

C

dz

dzd

2

0

2

0

4SI

I

K u

-

(2.24)

Where oi is the peak tip deflection due to vortex shedding in the ith mode of

vibration in m, CL = 0.16, H is the height in meters, Ksi is the damping parameter for the

ith more of vibration, Sn

strouhal number = 0.2 and zi

is the normalized mode shape.

Calculations of oscillation calculated using this formula are acceptable till 4

percent of the effective diameter. For values more that this the resultant is amplified

using a given formula.

Once this value is obtained the sectional shear force Fzoi and Bending moment

Mzoi at any height zo for the ith mode of vibration, as obtained as follows.

H

zo

ziziozoidzmfF IKS 21

24 (2.25)

H

zo

ziziozoidzmfF IKS 21

24 (2.26)

Where fi is the natural frequency in the ith mode of vibration and mz is the mass

per unit length of the chimney at section z in kg/m.

The mass damping factor Ksi required for the earlier equation is calculated using

the formula

2

2

d

mK sei

is V

G (2.27)

mei is the equivalent mass per unit length in kg/m in the ith mode of vibration, s

= 2, and = 0.016 (structural damping factor), is the mass density of air taken as 1.2

kg/m3 and d is the effective diameter taken as average diameter over the top 1/3 height

of the chimney in m.

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The equivalent mass per unit length in the ith mode of vibration can be calculated

using the formula given below. It is basically dependant on the amount of mass that is

available given the mode shape.

H

zi

H

ziz

ei

dz

dzm

m

0

2

0

2

I

I

(2.28)

The oscillation is caused by the wind. The mode in which the chimney vibrates is

decided by the wind speed. Higher modes need a higher wind speed for excitation.

Hence it is possible to know the wind velocities that causes shedding in the i th mode. It is

done with the help of the following equation.

n

criS

dfV 1 (2.29)

Since higher wind speeds are required to excite higher modes of vibration, it is

not necessary to consider all the modes of vibration for the purpose of design. All modes

which can be excited up to wind speeds of 10 percent above the maximum expected at

the height of the effective diameter shall be considered for subsequent analysis. If the

critical winds for any mode of vibration, exceeds the limits specified earlier, the code

allows the assumption that the problem of vortex excited resonance will not be a design

criteria for that and higher modes. In these cases across-wind analysis may not be

required.

The across-wind analysis using the random response method is also specified by

the code. The relevant expressions are given for chimneys of two types those with

little or no taper and those with significant taper. Taper is defined as

H

ddtaper

topav )(2 (2.30)

When the value of the taper is less than 1 in 50 (or 2 percent) the chimney is said

to have little taper.

For chimneys with little or no taper, the expression to calculate the modal

response at critical wind speed as given in equation 2.24 earlier

16


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ei

azi

ein

L

oi

m

dkdz

H

m

Ld

S

HdC

22

2

22

1

1

)2(225.1

VEI

SV

S

I

K

-

u

(2.31)

Where the RMS lift coefficient is taken as 0.12, correlation length in diameters is

taken as 1.0 and the aerodynamic damping coefficient is taken as 0.5.

Chimneys that are significantly tapered have the following equation

H

ei

aziziei

zeizeL

oi

m

dkdmS

tLHdC

0

2222

14

2

2

VEIS

SIIVK (2.32)

Where zei is the height in m at which a given expression is maximum in the ith

mode of vibration. The term in the expression is the power law exponent which was

discussed earlier with respect to the wind profiles. The value of this depends on the

Terrain Category and varies from 0.10 to 0.34.

The critical wind speed for exciting the mode of vibration is determined by the

equation.

n

ize

criS

dfV

1 (2.33)

Calculations begin by first taking zei =H and progressively decreasing till a

maximum in oi is observed. Also if the required velocity for excitation in any mode is

greater than the maximum velocity, the chimney will not be assumed to experience

much across-wind loads in that and higher modes. If this applies to the first mode of

vibration itself then the chimney has negligible across-wind loads.

The ACI code considers the across-wind loads due to vortex shedding for in the

design of chimneys when the critical velocity is between 0.5 and 1.3 Vzcr. Across-wind

loads are not considered outside this range.

Te critical velocity is calculated using the function.

t

cr

S

ufdV

)( (2.34)

17


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Where the St is the Strouhal number and is calculated using

)(log206.0333.025.0

udhS et (2.35)

d(u) is the mean outside diameter of the upper 1/3 of the chimney in feet, and h is

the height above the ground level.

The peak base moment at the critical velocity if determined by the equation.

x

x

E

p

as

crLSa

Cud

h

LShudV

aCS

g

GM

)(

2

4)(

2

22

EESU

(2.36)

Ma is evaluated over a range of wind speeds in the specified range of 0.5 to 1.3

Vcr to determine the maximum response. For values of velocity greater than Vcr the

value of Ma is multiplied with

4.1

1

)(

)(

3

44.1

-

cr

cr

zV

zVV(2.37)

The values of the various terms are given in the code including the peak factor,

mode shape factor and specific gravity of air.

The code also gives a formula for the calculation of the time period in the second

mode of vibration, although the final design needs a dynamic analysis. The values

obtained from the STRAP program were used in this calculations.

The results of the analysis are given below

18


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0

50

100

150

200

250

300

-400 -200 0 200 400 600 800

Mode 1 Mode 2

Figure 2.5 IS Simplified method & Figure 2.6 IS Random Response Method

0

50

100

150

200

250

300

-1000 -500 0 500 1000 1500

Mode 1 Mode 2

19


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The first graph refers to the result of the IS simplified method, whereas the

second graph refers to the IS Random response method.

As can be seen from the graph the moments in the first mode of vibration arevery similar for both the methods of calculation, whereas the moments for the second

mode of vibration vary a lot. The moments obtained from the Random response method

are almost double that obtained using the simplified method. In fact the Random

response method given higher moments for the second mode of vibration and lower

moments for the first mode of vibration, as compared to the simplified method.

The base moments as calculated using the ACI method are given below

(All values MNm) Across-wind Along-wind Gust Factor Max Moment

Mode 1 125.46 432.98 1.8854 825.922

Mode 2 98.86 432.98 1.592 696.56

Table 2.4 Base Moments (ACI)

It is seen that the values obtained using the ACI method are very small as

compared to the IS method. This is especially true of the across-wind loads.

2.2. Variation of moments with change in H by D ratioAn analysis was done to find the change in across-wind loads with change in

Height to Base diameter ratio.

For the purpose of the Analysis, Chimneys with the following parameters were

used

Height : 250 m

Height to Base diameter Ratio : 7, 9, 11, 12, 13, 15, 17

Top diameter to Base diameter Ratio : 0.6Base diameter to Base thickness Ratio : 35

Top thickness to Base thickness Ratio : 0.4675

The following methods were employed for the same

1. IS Approximate Method

2. IS Random Response Method

3. ACI 95 Method (Also CICIND approved)

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Estimation of Free Vibration parameters like the mode-shapes the free

frequency and the Weight data for the calculations were calculated by modeling thechimney in STRAP. The modeling was done with the chimney broken down into 32

elements. Vibration Analysis was done for modes 1 to 5 but only the first two were

required for the purpose of Moment calculations.

2.2.5 Conclusions from the variational analysis

x The Across-Wind Moments were inversely proportional to the H by D Ratio.The Moments consistently increased with fall in the H/D Ratio for all methods of

estimation.

x The Approximate method of the IS code gave consistently higher moments as

compared to the Random Response Method for vibrations in the first mode.

x The Approximate method of the IS code gave consistently lower moments as

compared to the Random Response Method for vibrations in the second mode.

x The IS method gave higher moments in the second mode of vibration as

compared to the first mode in both its methods.

x The ACI method gave very small values as compared to the IS methods for the

base moment in all cases

x Anomalously the moments in the second mode were lower in the ACI method as

compared to those in the first mode.

All relevant Data can be found in the subsequent pages. It may be noted that the

higher moment curves correspond to lower H/D ratio.

21


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0

5

10

15

20

25

30

35

0 2000 4000 6000

Figure 2.7 IS Approximate Method Mode 1

0

5

10

15

20

25

30

35

-10000 -5000 0 5000 10000

Figure 2.8 IS Approximate Method Mode 2

22


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0

5

10

15

20

25

30

35

0 1000 2000 3000 4000 5000

Figure 2.9 IS Random Response Method Mode 1

0

5

10

15

20

25

30

35

-20000 -10000 0 10000 20000 30000

Figure 2.10 IS Random Response Method Mode 2

23


35/84

H/d 7 9 11 12 13 15 17

Mode 1 641.15 340.566 204.425 144.783 114.783 77.271 55.244

Mode 2 411.482 225.483 142.764 107.867 87.404 59.786 42.523

Table 2.5 ACI Method (all modes)

Conclusion

The wind loads form the major sources for moments on Tall free standing

structures like chimneys. We have looked at the two kinds on wind-loads that act on

chimneys and also have presented the calculations for a standard chimney.

24


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Chapter 3. Estimation of Earthquake Load Effects

3.1 Introduction

Seismic action on chimneys forms an additional source of natural loads on the

chimney.

Seismic action or the earthquake is a short and strong upheaval of the ground.

This naturally is the cause for loads on any structure. Any structure under seismic loading

is subjected to cyclical loading for a short period of time.

An earthquake is described by its intensity and it epicenter.

The intensity of and earthquake at a place is a measure of the degree of shaking

caused during the earthquake and thus characterizes the effect of the earthquake. Most of

the study of earthquakes up to the beginning of the twentieth century dealt with the

effects of earthquakes and to quantitatively describe these effects a number of intensity

scales were introduced. Initially there was the Rossi-Forel scale that had ten divisions. In

1888 Mercalli proposed a scale with 12 subdivisions to permit a clear distinction in

shocks of extreme intensity. After a number of changes the Modified Mercalli scale or

simply the MM scale is generally used by engineers today. Another revision made in

1956 to the MM scale by Richter is also in use.

The focus is the source for the propagation of seismic waves. It is also called the

hypocenter. The depth of the focus from the surface of the earth directly above is referred

to as the focal depth. The point on the earths surface directly above the focus is known

as the epicenter.

The structure experiences cyclic loading during the process of seismic action.

This causes energy to build up in the system leading to its collapse. The friction with air,friction between particles that constitute the structure, friction at junctions of structural

elements, yielding of the structural material and other processes of energy dissipation

depress the amplitude of motion of a vibrating structure and the vibrations die out in

course of time. When such internal and or external friction fully dissipates the energy of

the structural system during its motion from a displaced position to its initial position of

rest, inhibiting oscillations of the structure, the structure is said to be critically damped.

25


37/84

Thus the damping beyond which motion will not be oscillatory is called critical

damping.

The effect of energy dissipation in reducing successive amplitude of vibrations ofa structure from the position of static equilibrium is called damping and is expressed as a

percentage of critical damping.

There are other terms that are important with respect to seismic analysis. During

earthquakes there occurs a sate in saturated cohesion less soil where in the effective shear

strength is reduced to a negligible value, for all engineering purposes. Un this condition

the soil tends to behave like a fluid mass.

A system is said to be vibrating in its normal mode or principal mode when all its

masses attain maximum values of displacement simultaneously and they also pass

through the equilibrium positions simultaneously. When a system is vibrating in its

normal mode, the amplitude of the masses at any particular time expressed as a ratio of

the amplitude of one of the masses is known as the mode shape coefficient.

During an earthquake ground vibrated (moves) in all directions. The horizontal

component of the ground motion is generally more intense than that of the vertical

components during string earthquakes. The ground motion is generally random in nature

and generally the random peaks of various directions may not occur simultaneously.

Hence for design purposes, at one time, it is assumed that only the horizontal component

acts in any one direction. All structures are designed to withstand their own weight. This

could be deemed as though a vertical acceleration of 1g is applied to the various masses

of the system. Since the design vertical forces proposed in the codes are small as

compared to the acceleration of 1 gravity, the same emphasis has not been given to the

vertical forces as compared to the horizontal forces. However for structures where

stability is a criterion it may become necessary to take into account these vertical forces.

3.2 Estimation of loads

The seismic action is described by means of a standardized acceleration response

spectrum. The CICIND code suggests a general response spectra. The response spectra is

a relation between the maximum effective peak ground acceleration at the location of the

26


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chimney. This is in relation with the natural time period of the structure and the soil type

existing at the site.

The movement of the chimney is found by calculating the first few mode shapesby modal analysis of the chimney. The result of such a modal analysis will yield the

values for the deflection, the shear force and the moment.

The modal analysis can determine the functions of the deflection, shear and the

moment only up to a constant factor. Thus if the mode shape calculated is known, then a

constant times the mode shape too is a possible solution.

Hence the actual value of the shear force or the bending moment is found by

multiplying the normalized response with a scaling factor.

Hence if u is the value of the normalized mode shape then the true mode shape is

given by

iii Nuu (3.1)

Where they refer to the ith mode of vibration, and Ni is the scaling factor. The

scaling factor is determined by the following equation.

)(4 2

2

isii Ta

TpNi

S (3.2)

The as is the response function described earlier. The value of pi is obtained from

h

i

h

i

i

dzzmzu

dzzmzu

p

0

2

0

)()(

)()(

(3.3)

The code also assumes the vertical movements to result in a value of resultants

that are 0.3 times the horizontal forces.

The ACI code also assumes the vertical component to be negligible with respect

to the horizontal one. The code also suggests the spectral values for the values of

maximum ground acceleration.

The following calculations are based on the IS code. The code used is the

IS:1893-1975.

27


39/84

Since the earthquakes occur without any warning, it is very necessary to avoid

construction practices that cause sudden failure or brittle failure. The current philosophy

relies heavily on the action of members to absorb all the vibrational energy resulting fromstrong ground motion by designing the member to behave in a ductile manner. In this

manner even if an earthquake occurs that is stronger than that which has been foreseen,

total collapse of the building can be avoided.

Earthquake resistant designs are generally performed by pseudo-static analysis,

the earthquake loads on the foundations are considered as static loads and hence capable

of producing settlement as dead loads. Therefore as the footings are generally designed

for equal stresses under them, the footings for exterior columns will have to be made

wider. Permissible increase in safe bearing pressure will have to depend in the soil-

foundation system. Where small settlements are likely to occur larger increase can be

allowed and vice versa.

3.2.1 Design seismic coefficients for different zones

The force attracted by any structure during an earthquake is dynamic in nature

and is a function of the ground motion and the properties of the structure itself. the

dominant effect is equivalent to a horizontal force varying over the height of the

structure. Therefore the assumption of a uniform force to be applied along one axis at a

time is an oversimplification which can be justified for reasons of saving effort in

dynamic analysis. However a large number of structures designed on this basis have

withstood earthquake shocks in the past. This is a justification of a uniform seismic

coefficient in seismic design. In the code, therefore, it is considered adequate to provide

uniform seismic coefficients to ordinary structures.

The IS code suggests two methods for the purpose of evaluation of the earthquake

loads. This is similar to the two methods suggested for the calculation of across-wind

loads. Both methods calculate the design value of the horizontal coefficient.

Seismic coefficient method

The value of the horizontal seismic design coefficient shall be calculated using the

following expression.

28


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2929

(3.4)0DED Ih

Where is a coefficient depending on the soil type. This value varies between 1.0 and

1.5.

I is the importance factor.

0 is the basic horizontal seismic coefficient.

The response spectrum method

The response acceleration is first obtained for the natural time period and

damping of the structure and the design value of horizontal seismic coefficient is

computed using the following expression.

g

SIF ah 0ED (3.5)

Here

F0 is a seismic zone factor.

Sa/g is the average acceleration coefficient depending on the natural period and

damping of the structure.

3.3 Calculations for a typical case

The calculation of the earthquake load for a typical chimney is given below. The

assumptions made are also specified.

The weight data for the case has been taken from the STRAP model of the

chimney.

Period of vibration

Diameter of the base = 22.72 m

Base Thickness = 0.649m

Inner diameter at the base is 21.422m

Area of cross section at the base is

(3.6)

22

4 inoutddA

S


41/84

A = 45.0 m2

The moment of inertia at the base is calculated by

4464

inout ddI S (3.7)

The value of I = 2742.5 m4

Radius of gyration r is given by

A

Ir (3.8)

r = 7.806

Hence the slenderness ration l/r is given by

02.32r

l(3.9)

The coefficient CT

(3.10)822.57TC

Weight of the chimney

(3.11)US tmeanTHDWt

Weight = 17495583 kg

The period of vibration is now given by

EAg

hWCT tT

' (3.12)

Substituting the values the value of T = 125.6

Design seismic coefficient

Using the Response Spectrum method and the equation **

ah = 0.03975

the value assumed are

= 1.0 (assuming a hard/medium soils)

I = 1.0 (importance factor)

30


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F0 = 0.25 (assuming the chimney to be in the zone IV)

Shear force and Bending momentsThe design shear force at a distance of X from the top is given by

2

'

'

3

2

'

'

3

5

h

X

h

XWCV thVD

(3.13)

Where the value of CV has been found to be 0.2 for the very large time period

obtained. Varying the value of X from 0 to 250 the profile of the shear force has been

calculated.

0

50

100

150

200

250

300

0 500 1000 1500

kN

Figure 3.1 Shear force due to seismic loads

The bending moment can be calculated using the formula

42

1

'

'4.0

'

'6.0

h

X

h

XhWM thD (3.14)

31


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Again the value of X is varied and the expression evaluated. The resultant graph

is given below.

0

50

100

150

200

250

300

0 200 400 600 800 1000

MNm

Figure 3.2 Bending Moment due to seismic loads

As can be seen from the graph, the maximum moment at the base of the chimney

is about 800 MNm.

3.4 Conclusions

The reasons and assumptions involved in the evaluation of earthquake loads have

been studied. The codal provisions for the calculation of the same have been understood.

A sample calculation has been done to calculate the shear force and bending moment

caused due to earthquake loading on chimneys. The loads in this case have been found to

be significantly lower that those obtained in the wind analysis. Hence earthquake loads

do not normally form the main loads to be considered for design.

32


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3333

Chapter 4. Estimation of Temperature Load Effects

4.1 Introduction

In walls of Reinforced Concrete chimney, stresses are developed due to the

temperature difference between the inner and the outer surface of the walls. This

temperature difference from inside to outside tends to expand the inner surface relative to

the outer one. Due to the monolithic action of the entire wall, differential expansion is not

possible and hence equal expansion takes place so that the shell is compressed on its

inside surface and pulled on its outside surface. As a whole there is an average increase in

length of the chimney due to the temperature gradient.

Various codes given different methods for the evaluation of the resultant

temperature stresses. The CICIND code does not explicitly give equations for the

evaluation of these stresses. Instead it asks the designer to account for them assuming the

shell wall to be a straight Reinforced Concrete wall.

The ACI code gives a code that is shortly discussed. There is another method that

is discussed by the book Advanced Reinforced Concrete Design by Dr. N.Krishna Raju

which will be used to calculate the stresses on a typical cross section.

The temperature stresses are of two different types. The stresses that occur in the

vertical part of the cross section and the stresses that occur in the horizontal part of the

cross section. Also calculations must be performed for the steel on the inside face as well

as the outside face of the chimney.

The ACI code gives the following equations to calculate the maximum vertical

stresses occurring in steel at the inside of the chimney due to temperature difference.

Note that fCTV refers to the concrete stresses and the value fSTV refers to the stress insteel.

cxteCTV ETcf xxxD'' (4.1)

And

cxteSTV nETcf xxx )1('' 2JD (4.2)

Where the terms are explained below


45/84

3434

@

te = thermal coefficient of expansion of the concrete and of the reinforcing steel,

to be taken as 0.0000065 per deg F

Ec = modulus of elasticity of concretec is given by the equation

> @ > )1(211 2122

11 JJJUJUJU nnnc (4.3)

= ratio of the total area of the vertical outside face reinforcement to total area of

concrete chimney shell at the section under consideration

1 = ratio of the inside face vertical reinforcement area to the outside face vertical

reinforcement area.

2 = ratio of distance between inner surface of chimney shell and center line of

outer face vertical reinforcement to total shell thickness

n = Es/Ec (4.4)

Tx is the temperature gradient across the shell.

The code gives a number of formulas for the calculation of this gradient

depending on the type of shell. The shell type could be any of unlined chimneys, lined

chimneys with insulation completely filling the space between the lining and the shell,

lined chimneys with unventilated air space between the lining and the shell or lined

chimneys with ventilated space between the lining and the shell.

The equation for the unlined case is given

coo

ci

cc

ci

i

oi

cc

ci

dK

d

dC

td

K

TT

dC

tdTx

1

(4.5)

Where the factors are dependant on the cross section under consideration.

The terms Ko and Ki are the coefficients for transfer of heat. These can be

obtained from curves given by the code.

The maximum stress in the vertical steel fstv occurring at the outside face of the

chimney shell due to the temperature gradient can be computed using

cxieSTV ETcf xxx 'D (4.6)


46/84

3535

An additional kind of temperature stress that is taken into account by the ACI

code is the circumferential temperature stress. The equation for the evaluation of the

same is

(4.7)cxieCTC ETcf xxx ''' D

and the same for steel is

(4.8)sxieSTC ETcf xxx )''('' 2JD

4.2 Equations for evaluation of stresses

The following is a derivation of the equations for the temperature stresses.

Assume that

To is the temperature difference between inside and outside with a linear

temperature gradient.

is the coefficient of expansion of steel and concrete.

e is the strain difference in temperature

m is the modular ratio

ts

is the area of reinforcement per unit width

tc is the area of concrete per unit width

ct is the stress in concrete due to temperature

st is the stress in steel due to temperature

p is (ts/tc)

k is the neutral axis depth constant.

Referring to the figures below and considering the force equilibrium we have

stcstscct pttkt VVV 2

1(4.9)

Which gives on solving for the stress in steel

k

kam

kt

ktatm

pct

c

ccct

ctst

VV

VV

2(4.10)


47/84

The following figures are a representation of the case

36

atc

tcAir Gap

Figure 4.1 Thermal Stresses

The expressions for stress in steel in turn give the following equation for the value

of k2

Wherein the value of k is

Lining

Temperature

Gradient To

ktc

stct

Net Strain in

Steel

(Tension) T- e

Te Net Strain in

Concrete

(Compression)

(4.11) kapmk 22

(4.12)222 mpmpampk


48/84

3737

Rise in temperature in reinforcement is

(4.13)Ta)1(

Free expansion of steel is(4.14)Ta D)1(

The tensile stress due to the difference between that due to strain e and due to

temperature rise (1-a)T

Hence the stress in steel is

(4.15) > @TaTkEsst DDV 11

or

(4.16))( kaTEsst DV

similarly stress in concrete is given by

(4.17)kTEcct DV

Stresses in horizontal reinforcement

At high temperatures, the inner surface of the chimney is prevented from

expansion and therefore gets compressed. The outer surface will expand more than the

natural expansion and will be in tension. Due to temperature stresses, generally the hoop

tries to expand and consequently tensile stresses develop in the hoop reinforcement.

Using the above figures and the following notation

ktc = position of the neutral axis

c = compressive strength in concrete

s = compressive strength in steel

As = area of hoop reinforcement per unit height

As = cross sectional area of steel


49/84

3838

The equations for the calculation of the stresses are given as

'

'''kkam cs VV (4.18)

(4.20)pmmppmak 222'

(4.21)> @ TaEm scs DVV ''

Knowing the value of k the stresses can be calculated.

Sample calculations

The following is a sample calculation for a simple 4000mm concrete Reinforced

Concrete wall. The derivation does not take into account the curvature of the shell

directly. Hence they can also be applied to any wall. Also the assumed thickness of the

wall is quite typical of the chimneys looked into so far.

tc = 4000

assuming a steel cover of about 100 mm

atc = 3900

hence a = 3900/4000 = 0.975

assuming a 1% steel reinforcement

p = 0.01

assume a temperature difference of 75oC

other values are

= 11*10-6 /oC

m = 11

Es = 210000

Ec = 19090.9

Calculating the value of k using equation 4.20

k = 0.366025

Hence the vertical stresses are calculated to be


50/84

c 5.765 N/mm2

s 105.5 N/mm2

Table 4.1 Vertical Stresses

The hoop stresses are calculated by solving the following equations

cs '3.18' VV (4.22)

And

(4.23)9.168'11' cs VV

Wherein the solutions are

c 5.764 N/mm2

s 105.49 N/mm2

Table 4.2 Hoop Stresses

4.3 Conclusions

The cause for the occurrence of thermal stresses in chimney shells were studied.Equations describing the phenomena were derived and stress resultants related. The

thermal stresses in the cross section were calculated.

39


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Chapter 5. Estimation of Design Resistance and

development of Interaction Curves

5.1 Introduction

This chapter deals with the calculation of the ultimate moment of resistance of the

Reinforced Concrete tubular section of the tower. There are many methods prescribed in

the codes for the purpose of estimation of the ultimate loads. These methods differ

primarily with regard to the model used to represent the stress strain curve of concrete in

compression.

The ultimate moment capacity of the tubular Reinforced Concrete section depends

on the normal compressive load that acts at that point. The interaction of this normal

force with the ultimate moment, corresponds particularly to the location of the neutral

axis which generally falls within the section for the high eccentricities in loading usually

encountered under extreme wind speeds.

The following are some of the assumptions commonly adopted for the purpose of

estimation.

1. Place sections remain plane after bending. This means that a linear strain

distribution is assumed at the cross section.

2. Extreme fibre stresses are computed at the center line of the concrete shell.

The mean radius is representative of all stresses.

3. The vertical reinforcing steel is replaced by an equivalent thin steel shell,

located at the mean radius.

4. The stress-strain relationship of steel is assumed to be elasto-plastic, and is

assumed to be identical in tension and compression.

5. Tensile stresses in concrete are ignored. The section is assumed to be fully

cracked in the tension region of the neutral axis.

In addition, the following are some requirements before the calculations can be

done.

x Stress-strain relationship of concrete in compression

x Limiting compressive strain in concrete

40


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x Limiting tensile strain in steel

x Modulus of elasticity of steel

The differences in the various codal methods are basically caused due to

dissimilarities in the above assumptions.

This paper calculates the design resistance using the standard stress-strain curve

for steel and a proposed stress-strain curve for concrete. This curve was proposed by Dr.

Devdas Menon in his Ph.D. thesis.

5.2 Characteristic Stress-Strain Curve for Steel

The stress-strain curve for steel is more or less standard and is used by all the

codal provisions. It is an idealized elasto-plastic relationship. The values to be assumed

are the Es (modulus of elasticity for steel) and the sml (limiting tensile strain in steel).

A diagrammatical representation of the Steel stress-strain curve is given below

fs Es = 200000MPa

41

Figure 5.1 Stress-strain curve (steel)

As has been indicted the value of

Es = 200,000 N/mm2

sml = 0.07 (as initially proposed by the ACI code)

The value for the limiting tensile strain is assumed for some codes to be a very

conservative 0.05. This is probably to take care of the excessive cracking in concrete on

the tension side. This however is not strictly called for at ultimate loads, in the limit state

s

fcyk

Es

smlsy

sml = 0.070


53/84

of collapse, since the crack control is checked for separately as part of the serviceability

requirements.

5.3 Characteristic stress-strain curve for concrete

Various codes give various stress-strain codes for concrete.

The ACI code for example employs the Hognestads curve, originally proposed

for eccentrically loaded columns. The curve has two parts. The first is a parabolic curve

and the second is a straight line that continues from the end of the parabolic curve that

represents the downward trend of the curve. It assumes a limiting strain under direct

compression of 0.002 and an ultimate strain in flexure of 0.003.On the other hand, the CICIND has a very elaborate curve. It is a parabolic-linear

curve that distinguishes between the effects of dynamic, short-term loading and static

long-term loading.

The curve that is used for the purpose of estimation of resistance and for the

purpose of generation of the interaction curves is a new curve. This curve has been

proposed taking into account the effect of tubular geometry and the effect of short-term

wind loading.

The limiting compressive strain in concrete cul corresponds to the maximum

value of the strain cu at the middle of the concrete shell thickness at the extremity of

compression. Since the shell is extremely thin in comparison to its very large diameter,

the distribution of stress across the thickness of the shell is almost uniform. The behavior

of thin walled chimneys is very different from the behavior of solid Reinforced Concrete

sections which can accommodate a large strain variation across the cross section.

Hence the value ofcul should not be as large as 0.003 as suggested by the codes.

Rather it must be restricted to a value usually specified under conditions of uniform

compression, that is cul = 0.002.

The CICIND code proposition of distinctively accounting for the dynamic short-

term loading effect of wind merits consideration. However the premises on which the

curve is based are questionable. It is, for example, observed that the wind loads are

extremely short-lasting, while the meteorological practice is to compile hourly mean

wind speeds. The values for the code are taken from practical tests where the loading was

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done by reversed cyclic bending. However since the dynamic nature of wind consists of

random velocity fluctuations about a mean, rather than complete change of direction in

short periods. Since the mean response to wind loading is fairly substantial and theoverall response is quasi-static in nature, the behavior is better approximated by

monotonic loading rather that reversed cyclic loading; the duration of the loading to be

considered is approximately 2 to 5 hours.

On the basis of the results of a large number of tests on eccentrically loaded

concrete cylinders under varying load conditions the following conclusions can be drawn

x The stress strain curve is parabolic rather than linear, even under the

short term loading under consideration.x If fcu = 0.85 fck is assumed then it is reasonable to assume an increase

of approximately 10% for relatively short time loading.

x The value of the ultimate compressive strength cul corresponding to

this peak may be assumed to be approximately 0.002 for both short-

term and long-term loading.

On the basis of the above discussion the following curve is assumed as the stress-

strain curve for concrete under compression. It employs a simple parabolic curve with a

limiting ultimate limiting strain of 0.002 and a value of fcu = (0.85 fck) CS. Here the term

CS is called the short term loading factor, having a value that depends on the normal

compression on the tower section; it is assumed to vary linearly between a maximum

value of (0.95/0.85) for normal load = 0 and to unity when the value of normal load is

maximum that is under pure compression.

The formula for the curve is given below

pcspc fCf 1 (5.1)

where

85.0

1.095.0max

-

N

N

Cs

(5.2)

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The curve is as shown in the following figure.

fcu 0.85 Cs fck

c1

cul

Figure 5.2 Stress-strain curve (Concrete)0.002

Design Stress-Strain Curve

The characteristic stress-strain curve refers to the actual characteristic values of

the stress-strain values. These are multiplied by the partial safety factors to get the design

curves. The values of the partial safety factors assumed are as follows

s = 1.15

c = 1.50

these design curves are used to calculate the design ultimate moment carrying

capacity of the Reinforced Concrete tubular section.

The codes also specify either the design or the characteristic curves. The CICIND

code for example specifies the design curves along with the characteristic curves whereas

the ACI method specifies the design curve which is to be multiplied with a resistance

factor of 0.8. The code does not recommend any design stress-strain curves.

5.4 Calculation of Ultimate moments

The ultimate moment carrying capacity Mu of tubular section, corresponding to

any given normal compression N is determined by solving the following equilibrium

equations.

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45

(5.3)

where Nc and Ns are the resultant normal forces obtained from the concrete and

steel stress blocks respectively. Muc and Mus denote the respective moments of the

concrete and steel blocks about the centerline.

The following diagram is a representation of the various components involved in

the estimation of the design interaction curves.

Figure 5.3 Chimney Cross section

The distribution of strains and the corresponding stresses are given in the below

Neutral Axis

sc NNN

usucu MMM (5.4)

0

Wind


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Neutral Axis

Strains

46

Figure 5.4 Stress and strain distributions

These diagrams are merely depictive. They do not show the actual values.

As can be seen from the diagrams, for a neutral axis there exists a strain

distribution. This strain distribution is linear because of the assumption we had made in

the starting of the chapter. This in turn determines the stresses in the concrete and steel

block. The summation of these stresses gives rise to the resistive strength of the

chimneys.

5.5 Interaction Curve

The interaction curve is a complete graphical representation of the design strength

of a Reinforced Concrete chimney. Each point on the curve corresponds to the design

strength values of N and Mu. That is to say that if the load of N were to be applied to the

Reinforced Concrete chimney with an increasing eccentricity then the value of the

eccentricity where this line would intersect with the interaction curve is given by

cu = 0.002

Concrete

Stresses

N

MuH

fcu

Steel

Stresses

-fsyk

fsyk

(5.5)


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The interaction curve is the failure envelope. Any point inside the curve is safe.

That is any combination of moment and compressive strength where the point lies withinthe curve will not cause failure of the Reinforced Concrete chimney.

In reality the loading is not done in this manner. Given values the moment and the

compressive stress, it should be possible to check whether the chimney cross section is

safe.

The magnitude of N determines the neutral axis. This location is specified by the

angle 0 in the equation and the diagram given above. On location of the neutral axis the

strain distribution is known. This can then be used to solve for the value of N and the

ultimate moment Mu. It is therefore obvious that the solution to the above set of equations

can be found as a closed form solution. This is because the location of the neutral axis is

required for the calculation of the normal force N, while the value of N is itself required

for the location of the neural axis.

For the purpose of developing the interaction curves the location the neutral axis

was assumed and the values of the normal force and the moment were calculated. The

neutral axis was then changed to calculate a new set of N and Mu. This was repeated to

get the interaction curves of N Vs Mu.

Not all locations of the neutral axes are realistically feasible, as will be seen in the

following discussion.

The following diagram depicts the variation of the strain profile with change in

the location of the neutral axis.

47

Figure 5.5 Strain profile variation

The maximum

compressive

strain in steel=0

=90 = maximum

Neutral axislocation not

possible


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As the angle that locates the neutral axis changes from 0 the location of the

neutral and hence the participation of steel in taking the load varies. This continues as

more and more participation of steel in tension occurs and the net compressive force onthe chimney reduces. At a particular value of the value of steel in tension effectively

nullifies the effect of the compression of the concrete block. Any increase in the value of

is not possible because it follows that the chimney in overall tension, which is not

possible.

Although the interaction curve is plotted between the value of N and Mu, in the

interest of greater flexibility, the interaction curve is rendered non dimensional by use of

the following relations

rtf

Nn

ck' (5.6)

trf

Mm

ck

u

2' (5.7)

Where r is the value of the radius of the section in consideration of the Reinforced

Concrete chimney, and t is the thickness of the section.

5.5.1 Family of interaction curves

Since we are using the non dimensional parameters m and n, the curves are no

longer applicable to one chimney alone. It is possible to plot a family of curves that vary

with respect to one parameter. Once the parameter value is known, it is possible to

calculate the corresponding value for any new chimney and then reuse these curves for

that particular chimney.

The parameter that was used for the purpose of generating a family of curves was

ck

syk

f

f

'U (5.8)

Where

is the percentage of steel

fsykand fckare the strengths of steel and concrete.

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A program was written in C++ that was used to that calculate the values of pairs

of values of n and m. The iteration was done by varying the value of the angle of the

neutral axis in incremental steps of 1 degree. Then the strain distribution for thatparticular neutral axis was evaluated. The total force contributed by the concrete and steel

sections was evaluated by integration. Then the value obtained was non-dimensionalised

using the factors as appropriate. This was continued till the value of the total normal force

evaluated to zero, signaling that the limit of the neutral axis was achieved. The program

listing is given in the appendix.

The interaction curve is given below.

Interaction Curves

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4

m

n

2.075

8.3

15.56

20.75

25.94

31.125

Figure 5.6 Interaction curves

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The family of curves is from the parametric variation of the term given earlier.

The values of the parameter for each of the curve is given in the chart. The curves have to

be read left to right. That is the first curve on the left refers to the value

075.2'

ck

syk

f

fU (5.9)

And so on.

The values of the terms utilized to arrive at the values are given below

fck fsyk (fsyk/fck)

0.2 40 415 2.0750.8 40 415 8.3

1.5 40 415 15.5625

1.5 30 415 20.75

1.5 24 415 25.94

1.5 20 415 31.125

Table 5.1 Values of the interaction curve parameter

From the table the ranges assumed for the values are also visible. The percentage

of steel is assumed from 0.2% to 1.5% which is the normal range. The value of f ck too is

assumed to be varying from 20 to 40, that is use of concrete of grades M20 to M40 has

been assumed.

The usage of these curves for the estimation of strength is shown in the chapter

Design and detailing of Example Chimney.

5.5.2 Derivation of equations used

The derivation of the equations for the calculation is given below.

50

(5.10)cks fCfcu '85.0

Where

Cs is the short term loading factor that varies linearly as explained earlier.

cul = 0.002


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The stress-strain curve for concrete is given below

-

2

11

2ccc

cupc

ff

H

H

H

H

J

(5.11)

The stress-strain curve for steel is given below

-

o

ddo

HH

H

H

HHHH

sy

sysys

fsyk

E

fs (5.12)

Where

ss

syk

syE

f

JH (5.13)

Let Nc and Ns refer to the compressive forces in the concrete and steel blocks

respectively. Similarly Mc and Ms refer to the moments in the two blocks. Then the

integration equations are

S

D

DHU

0

)()1(2 dfrtN pcc(5.14)

S

D

DDHU

0

)cos()()1(2 2 dftrM pcuc (5.15)

S

DHU

0

)(2 dfrtN ss (5.16)

S

DDHU

0

2 )cos()(2 dftrM sus (5.17)

But it is not necessary to calculate the value of the whole normal force or the

moment. It is only required to calculate the value of the non dimensional parameters.

Using the relations given in equation 5.6 and 5.7 we have

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S

D

DH

U

0

)('

)1(2

dffn pcckc (5.18)

S

D

DDHU

0

)cos()('

)1(2df

fm pc

ck

c(5.19)

S

DHU

0

)('

2df

fn s

ck

s(5.20)

S

DDHU

0

)cos()('

2 dff

m sck

s(5.21)

Note that 0 is the parameter for varying the location of the neutral axis.

These four equations form the basis for the calculation of the interaction curves

shown above.

5.6 Conclusions

The stress-strain curves of the steel and the special curve for concrete were

formed and justified. The ultimate strength equation was formulated. The interaction

curve between moment and compressive force was calculated and plotted. The necessary

equations for the same were also derived and listed.

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6. Designing and Detailing of example chimney

6.1 Introduction

In the earlier chapters about analysis of the various loads that are incident on a

chimney, a number of calculations have been performed on some typical chimneys.

Those results will be brought together towards the design of a sample chimney.

Then the detailing of such a chimney is also shown.

In addition the last part of the chapter deals with the design of the footing for the

chimney.

6.2 Design of a chimney

The following table gives the list of the various parameters of a chimney and their

typical values.

Name of parameter Practical range Typical value

Slenderness ratio h/Do 7-17 11

Taper ratio Dt/Do 0.3-1.0 0.6

Base diameter to thickness ratio

Db/tb

20-50 35

Mean, base thickness ratio tm/tb 0.3-0.8 0.55

Top mean thickness ratio tt/tm 0.7-1.0 0.85

Table 6.1 Chimney parameters

These values determine the section of the chimney which is given below with the

dimensions of the various parameters.

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13.6 m

Top Thickness

0.3035 m

250 m

Base Thickness

0.65 m

22.72 m

Figure 6.1 The chimney

Checking the viability of the cross section

Taking the values of the forces as follows, which have been calculated in the

earlier chapters. It may be noted that this calculation is for the worst case of the wind

load.

Moment = 1552.8 MNm

Axial force = 175 MN

Calculating the values of m and n to be used in the design charts, assuming

M30 concrete.

m = 3.089

n = 3.955

The parameter value for use in the design charts without the value of the steel

comes to

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Parameter = 13.83 (percentage of steel)

With 1% steel, using the curve with a parametric value of 15.56 the crosssection is safe.

The design

Fe415 steel

M30 concrete

Steel = 1%

Since the loads and other effects are totally reversible, the steel must be applied

equally on both faces of the chimney shell. Hence each face has 0.5 percent of the steel.

The detailing is done as follows and the figure is given later.

Using bars of 25mm diameter

Area of a meter length (circumferential) of the chimney = 6500mm2

Area of reinforcing bar = 490.9 mm2

Number of bars = 6.6

Spacing between the bars = 150 mm

Provide a cover of 75 mm on either face

This is the scheme to be followed for both the horizontal and the vertical

reinforcement at the base of the chimney. The changes to be done are given below

Curtailment

Since the chimney tapers with height, the area of concrete available decreases

along with the reinforcement requirement. Hence the vertical steel needs to be curtailed

in stages. The following scheme may be followed for the same.

Curtail 1 out of every 6 six bars at about a height of 120m. Curtail a second bar

out of the original six (now five) at a height of 200m.

The horizontal s

Documents

Analysis and Design of RC Chimney 120m